{"title":"一个涉及广义抛物柱面函数积的四重积分𝐷ᵥ(递减递减)𝐷ᵤ(递减递减):推导和求值","authors":"Robert Reynolds, Allan Stauffer","doi":"10.1090/bproc/126","DOIUrl":null,"url":null,"abstract":"<p>The aim of the present document is to evaluate a quadruple integral involving the product of the generalized Parabolic Cylinder functions <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D Subscript v Baseline left-parenthesis beta x right-parenthesis upper D Subscript u Baseline left-parenthesis alpha z right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>D</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>v</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>β<!-- β --></mml:mi>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n <mml:mi>D</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>u</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>α<!-- α --></mml:mi>\n <mml:mi>z</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">D_{v}(\\beta x)D_{u}(\\alpha z)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> expressed in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"18 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A quadruple integral involving the product of generalized parabolic cylinder functions 𝐷ᵥ(𝛽𝑥)𝐷ᵤ(𝛼𝑧): Derivation and evaluation\",\"authors\":\"Robert Reynolds, Allan Stauffer\",\"doi\":\"10.1090/bproc/126\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The aim of the present document is to evaluate a quadruple integral involving the product of the generalized Parabolic Cylinder functions <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D Subscript v Baseline left-parenthesis beta x right-parenthesis upper D Subscript u Baseline left-parenthesis alpha z right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>D</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>v</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>β<!-- β --></mml:mi>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:msub>\\n <mml:mi>D</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>u</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>α<!-- α --></mml:mi>\\n <mml:mi>z</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">D_{v}(\\\\beta x)D_{u}(\\\\alpha z)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> expressed in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"18 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-04-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/126\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/126","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
本文的目的是计算广义抛物柱面函数dv (β x) du (α z) D＿v({}\beta x){D＿u}(\alpha z)用Hurwitz-Lerch zeta函数表示的积的四重积分。特殊情况用基本常数来计算。这项工作的所有结果都是新的。
A quadruple integral involving the product of generalized parabolic cylinder functions 𝐷ᵥ(𝛽𝑥)𝐷ᵤ(𝛼𝑧): Derivation and evaluation
The aim of the present document is to evaluate a quadruple integral involving the product of the generalized Parabolic Cylinder functions Dv(βx)Du(αz)D_{v}(\beta x)D_{u}(\alpha z) expressed in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.
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